Problem 87 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate constant, the activation energy and the Arrhenius parameter of a chemical reaction at \(25^{\circ} \mathrm{C}\) are \(3.0 \times 10^{-4} \mathrm{~s}^{-1}, 104.4 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(6 \times 10^{14} \mathrm{~s}^{-1}\) respectively. The value of the rate constant as \(\mathrm{T} \longrightarrow \infty\) is (a) \(2.0 \times 10^{18} \mathrm{~s}^{-1}\) (b) \(6.0 \times 10^{14} \mathrm{~s}^{-1}\) (c) infinity (d) \(3.6 \times 10^{30} \mathrm{~s}^{-1}\)

Step-by-Step Solution

Answer

The correct answer is (b) \( 6.0 \times 10^{14} \mathrm{~s}^{-1} \).

Step 1: Understanding the Arrhenius Equation

The Arrhenius equation is given by \( k = A e^{-E_a / (RT)} \), where \( k \) is the rate constant, \( A \) is the Arrhenius parameter (or frequency factor), \( E_a \) is the activation energy, and \( R \) is the gas constant (8.314 J/mol·K). \( T \) is the temperature in Kelvin.

Step 2: Asymptotic Behavior of Temperature

We are asked to find the behavior of the rate constant \( k \) as \( T \to \infty \). As temperature increases to infinity, the term \( e^{-E_a / (RT)} \) will tend to 1, because the exponent \( -E_a / (RT) \) approaches 0.

Step 3: Limit of Rate Constant at Infinite Temperature

Given that the limit \( e^{-E_a / (RT)} \to 1 \) as \( T \to \infty \), the rate constant \( k \) approaches the Arrhenius parameter \( A \). Therefore, \( \lim_{T \to \infty} k = A = 6.0 \times 10^{14} \mathrm{~s}^{-1} \).

Step 4: Conclusion on Choice of Answer

With \( A = 6.0 \times 10^{14} \mathrm{~s}^{-1} \), the rate constant as \( T \to \infty \) is \( k = 6.0 \times 10^{14} \mathrm{~s}^{-1} \). Thus, the correct option is (b).

Key Concepts

Activation Energy

Activation energy is a crucial concept in chemical kinetics. This term refers to the minimum amount of energy needed for reactants to transform into products in a chemical reaction. It is often symbolized as \( E_a \) and measured in kJ/mol.
The activation energy serves as a barrier that must be overcome for a reaction to occur. Higher activation energy implies that more energy is required to start the reaction, making it slower at a given temperature. Conversely, a lower activation energy means the reaction can proceed more quickly.
When we apply the Arrhenius equation, \( k = A e^{-E_a / (RT)} \), the exponential term \( e^{-E_a / (RT)} \) accounts for the fraction of molecules that have enough energy to overcome this barrier. If the activation energy is high, the exponential term becomes smaller, and thus the rate constant \( k \) also becomes smaller, meaning the reaction proceeds slower. Understanding activation energy is key to controlling reaction speeds, which is crucial in fields like chemistry and biochemistry.

Rate Constant

The rate constant, denoted as \( k \), plays a significant role in determining the speed of a chemical reaction. It is a proportionality factor in the rate law that links the reaction rate to the concentrations of reactants.
The rate constant's value is influenced by several factors such as temperature and the presence of a catalyst. In the Arrhenius equation \( k = A e^{-E_a / (RT)} \), it is clear that \( k \) is directly dependent on both the activation energy \( E_a \) and the temperature \( T \). Increasing the temperature generally increases the value of \( k \), implying that the reaction rate would also increase. This is because higher temperatures provide more energy, allowing more molecules to surpass the activation energy barrier.
The units of the rate constant vary depending on the order of the reaction, but they often include time \((s^{-1})\) since many reactions are measured based on how long they take to complete. Understanding the rate constant is essential for predicting how quickly a reaction will proceed under specific conditions.

Frequency Factor

The frequency factor, also known as the pre-exponential factor, is denoted by \( A \) in the Arrhenius equation. It represents the frequency of collisions between reactant molecules that would lead to a reaction.
Typically, \( A \) is a constant for a given reaction at a specific temperature. It encompasses both the frequency of collisions and the probability that those collisions result in a reaction. The frequency factor hints at the orientation and spatial distribution needed for the reactants to successfully transform into products. Low values of \( A \) suggest that effective collisions are rare, while high values indicate a higher likelihood that reactants will collide in the proper orientation to react.
The frequency factor is a crucial part of the kinetic model, providing insights into how molecular dynamics affect the reaction rate. It also helps to estimate the rate constant \( k \) at various temperatures by keeping track of collision dynamics apart from energy considerations. The concept underscores the complexity of reactions and emphasizes the role of molecular interactions in chemical processes.